Quest HW 8 - roberts (eer474) Quest HW 8 seckin (56425) 1...

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Unformatted text preview: roberts (eer474) Quest HW 8 seckin (56425) 1 This print-out should have 16 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Find the linearization of f ( x ) = 1 3 + x at x = 0. 1. L ( x ) = 1 3- 1 3 x 2. L ( x ) = 1 3 parenleftBig 1- 1 6 x parenrightBig correct 3. L ( x ) = 1 3 parenleftBig 1- 1 3 x parenrightBig 4. L ( x ) = 1 3 parenleftBig 1 + 1 6 x parenrightBig 5. L ( x ) = 1 3 parenleftBig 1 + 1 6 x parenrightBig 6. L ( x ) = 1 3 + 1 3 x Explanation: The linearization of f is the function L ( x ) = f (0) + f (0) x . But for the function f ( x ) = 1 3 + x = (3 + x ) 1 / 2 , the Chain Rule ensures that f ( x ) =- 1 2 (3 + x ) 3 / 2 . Consequently, f (0) = 1 3 , f (0) =- 1 6 3 , and so L ( x ) = 1 3 parenleftBig 1- 1 6 x parenrightBig . 002 10.0 points Use linear appproximation to estimate the value of 17 1 / 4 . ( Hint : (16) 1 / 4 = 2.) 1. 17 1 / 4 65 32 correct 2. 17 1 / 4 2 3. 17 1 / 4 31 16 4. 17 1 / 4 33 16 5. 17 1 / 4 63 32 Explanation: Set f ( x ) = x 1 / 4 , so that f (16) = 2 as the hint indicates. Then df dx = 1 4 x 3 / 4 . By differentials, therefore, we see that f ( a + x )- f ( a ) df dx vextendsingle vextendsingle vextendsingle x = a x = x 4 a 3 / 4 . Thus, with a = 16 and x = 1, 17 1 / 4- 2 = 1 32 . Consequently, 17 1 / 4 65 32 . 003 10.0 points If f is the function defined on [- 4 , 4] by f ( x ) = x + | x | - 4 , which of the following properties does f have? A. Absolute minimum at x = 0 . B. Differentiable at x = 0 . roberts (eer474) Quest HW 8 seckin (56425) 2 1. neither of them 2. both of them 3. A only correct 4. B only Explanation: Since | x | = braceleftbigg x , x ,- x , x < , we see that f ( x ) = x + | x |- 4 = braceleftbigg 2 x- 4 , x ,- 4 , x < . Thus on [- 4 , 4] the graph of f is 2 4- 2- 4 2 4- 2- 4 Consequently: A. True: by inspection....
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This note was uploaded on 11/08/2010 for the course MATH 408K taught by Professor Gualdani during the Spring '09 term at University of Texas at Austin.

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Quest HW 8 - roberts (eer474) Quest HW 8 seckin (56425) 1...

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